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A three-layer Hele-Shaw problem driven by a sink

Published online by Cambridge University Press:  31 October 2024

Meng Zhao Open the ORCID record for Meng Zhao [Opens in a new window] ,
Amlan K. Barua ,
John S. Lowengrub ,
Wenjun Ying  and
Shuwang Li
Meng Zhao*
Affiliation:
Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, PR China
Amlan K. Barua
Affiliation:
Department of Mathematics, IIT Dharwad, Dharwad, Karnataka 580011, India
John S. Lowengrub
Affiliation:
Department of Mathematics, University of California-Irvine, CA 92521, USA
Wenjun Ying
Affiliation:
School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, PR China
Shuwang Li*
Affiliation:
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
*
†Email addresses for correspondence: mzhao9@hust.edu.cn, sli15@iit.edu
†Email addresses for correspondence: mzhao9@hust.edu.cn, sli15@iit.edu
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Abstract

In this paper, we investigate a sink-driven three-layer flow in a radial Hele-Shaw cell. The three fluids are of different viscosities, with one fluid occupying an annulus-like domain, forming two interfaces with the other two fluids. Using a boundary integral method and a semi-implicit time stepping scheme, we alleviate the numerical stiffness in updating the interfaces and achieve spectral accuracy in space. The interaction between the two interfaces introduces novel dynamics leading to rich pattern formation phenomena, manifested by two typical events: either one of the two interfaces reaches the sink faster than the other (forming cusp-like morphology), or they come very close to each other (suggesting a possibility of interface merging). In particular, the inner interface can be wrapped by the other to have both scenarios. We find that multiple parameters contribute to the dynamics, including the width of the annular region, the location of the sink, and the mobilities of the fluids.

JFM classification

Low-Reynolds-number flows: Hele-Shaw flows Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Boundary integral methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 998 , 10 November 2024 , A35
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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